Download A Definition of Boundary Values of Solutions of Partial Differential

Publ. RIMS, Kyoto Univ.
19 (1983), 1203-1230
A Definition of Boundary Values of Solutions of
Partial Differential Equations
with Regular Singularities
By
Toshio OSHIMA*
§ 0.
Introduction
In a symposium on hyperfunctions and partial differential equations held at
Research Institute for Mathematical Science, Okamoto introduced the following
Helgason's conjecture: Any simultaneous eigenfunction of the invariant differential operators of a Riemannian symmetric space of non-compact type has a
Poisson integral representation of a hyperfunction on its maximal boundary.
There had been several affirmative results in some cases. But the method used
there was hard to apply to general cases. On the other hand, taking this introduction by Okamoto, we constructed the concept of boundary value problems
for differential equations with regular singularities in [5] to solve this conjecture
and then it was completely solved in [4].
Recently the conjecture, which is now solved, reveals many important
applications in the theory of unitary representations. But the method in [5]
is not always easy to be understood by every person. In this note we introduce
another definition of the boundary values in an elementary way. Also we give
several results concerning the definition, which are sufficient to solve "the
conjecture" and moreover Corollary 5.5 in [6] which determines the image of
the Poisson transformation of Schwartz's distributions on the boundary. Such
applications of this note will appear in another paper.
§ 1.
Differential Operators with Regular Singularities
As usual, Z is the ring of integers, N the set of non-negative integers, Z+
Received May 19, 1983.
* Department of Mathematics, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo
113, Japan.
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TOSHIO OSHIMA
the set of strictly positive integers, Q (resp. R, resp. C) the field of rational
(resp. real, resp. complex) numbers and R+ the multiplicative group of strictly
positive real numbers.
Let n be a non-negative integer, X a domain in C1+w (or a (1 -f n)-dimensional
complex manifold) and 7 a 1-codimensional closed submanifold of X. Then
for any point p of X there exists a local coordinate system (r, x) = (f, x l 9 - - - , x,,)
defined in a neighborhood U of p such that
Unless otherwise stated, we use the above local coordinate system (f, x) and do
not write U. Moreover in many cases in this section we may assume X = U
because our concerns are reduced to essentially local problems.
Let 0(X) (resp. 0(7)) denote the ring of holomorphic functions on X
(resp. 7) and &(X) the ring of holomorphic differential operators on X which
are of finite order. For simplicity we use the following notation
For a multi-index a = (a l s ---, a,7)e JV" we denote
x
=x
...xn>
Any P = P(t, x; Dt, Dx) in &(X) has the form
(1-1)
P=
Z
PtJt
with a suitable m e ^V. Its symbol crm(P) of order m is defined by
(1.2)
t7 m (P)(r,x,T, 0=
Z
A,.C>*W-
Here ^ = (^ 1 9 -.., 4'J. Then am(P) belongs to the polynomial ring 0(X)[x9 f] of
(T, £) with coefficients in 0(X). If crm(P)^0, m is called the order of P (or
ord (P)) and crm(P) is called the principal symbol of P (or cr(P)). Furthermore
we denote by @(m)(X) the module of differential operators in @(X) whose order
are at most m.
DEFINITION OF BOUNDARY VALUES
1205
The following lemma is easy but will be frequently used.
Lemma 1.1. 1) rs0ts = 0 + s (seC),
2) tkDk = 0(0-l)-"(0-k+l)
(keN-{Q})>
3)
4) tD, =
Proof.
t'D,.9Le.e =
- if
t = t".
1) For a function $(t, x)
2) We will prove the equation by the induction on k. Since it is clear
when k = l, we have by the induction hypothesis
= t0r1-t(0-l)rl—t(&-k+l)rl-tDt
3) The equation reduces to 1) when k = l . Using the induction on k,
we have
4) Since
T
= ktfk'\Dtf = ktfk-1Dt and t'Dt, = k't'kDt = ktDT.
Q.E.D.
Now we prepare the following lemma to define differential operators with
regular singularities.
Lemma 1.2. For a differential
ditions a) and b) are equivalent:
a) P has the following form:
(1.3)
the following con-
P(t, x : Dt, Dx) = Z a/*)<9' + «S(f, x ; 0, /),) .
j=o
b)
(1.4)
(1.5)
operator Pe&(m)(X)
T/ierg exisfs a(x, s) e <?( Y)[s] sue/? f/iaf /or any 0 6 (P(X) and s e C
(r'P^
(r'P^
1206
TOSHIO OSHIMA
a)=>b): Assume P has the form as in a). Since t~s0Jts
\ we have
Proof.
rsP(t, x; Dt, DJts=
j=o
=
dj(x)rs0Jts + t -rsQ(t, x; 0, Dx)ts
aj(x)(0 +Sy + rQ(r, x; 0 + s, Dx)
o
with a suitable £(f, x; £>„ DJe&(X).
Hence b) is clear by putting a(x, 5)
= i a/^y •
J=0
b)^>a): Assume b). Putting
R(t, x; Dt, Dx) = P(t, x; Dt, Dx)-a(x, 0),
we have only to prove that R has the form R(t, x ; Dt, Dx) = tQ(t, x ; 0, Dx). Since
for any $ e &(X)
= 0,
s
s
we have (r Rt )0(X)<=:0(X)i
for any seC. Next we put
m
R(t,x;Dt9DJ=?,Rtt9X;Dx)Di
j=o
with Rj e ^m~j\X).
Then for any <j)(x)
m
rsRt, x;
t, x;
x;
Putting s = 0, 1, 2,---, we have
rJRj(t, x ; Dx)<^(x) e 0(JQ* for any
Hence Rj(t, x; Dx) = tJ + 1QJ(t, x; Dx) with suitable QjE @(X) and then
=/
DEFINITION OF BOUNDARY VALUES
This means a).
1207
Q.E.D.
Definition 1.3. We denote by &$(X) the subspace of &(X) formed by the
differential operators which satisfy the equivalent conditions in Lemma 1.2 and
for P e &$(X) we denote by cr*(P) the polynomial a(x, s) (= Ea/x)^ e 0(Y)M)
using the notation in the lemma. Here we remark that if we put 0 = 1, it is
clear that a(x, s) is uniquely determined. Moreover the condition b) says that
is a subalgebra of @(X) and that the map
defines a C-algebra homomorphism. Now we denote by &y(X) the subalgebra
of @$(X) generated by ®$(X) fl @™(X) ( = {Pe ®v\X)\ ff1(P)|y = 0}). Then
a differential operator P of order m is said to have regular singularities (or
R.S. for short) along Y if P belongs to @?(X) and if G*(P)(x9 s) is of just degree
m for any xeY. In the above if P belongs to @*(X) in place of @?(X) we say
that P has R.S. along Y in a weak sense. Jn both cases o"#(P) is called the
indicial polynomial of P and the roots of the equation a%(P)(x, s) = 0, which
we denote by s^x),--, sm(x), are called the characteristic exponents of P.
Similarly the differential equation Pw = 0 is said to have R.S. (or R.S. in a weak
sense) along 7 if P is so. In this case the equation o-^(P) = 0 is called the indicial
equation of the differential equation and the roots s^x),---, sm(x) are called its
characteristic exponents.
Proposition 1.4. Let P be a differential
following three conditions are equivalent:
a) P has R.S. along Y.
b) P has the form
(1.6)
operator of order m.
P= Z tJ'Pj(t, x; Df, Dx) with
Then the
Pj e ®
and <rm(Pm)(0, x; 1, 0)^0 for any xe Y.
c) P has the form
(1.7)
P = P(t9x;e,tDJ
c
m,o(°> x )^° /or
Proof.
an
J xeY, where (tDJC)a = (tDJCl)ai---(rDJCn)a" .
It is clear that b) and c) are equivalent (cf. Lemma 1.1. 2)). Since
1208
TOSHIO OSHIMA
is
the subalgebra of 9(X) generated by <9, tDxl,—, tDXn and &(X),
the relations
[0, 0] e 6(X\
[tDXi, 0] G <9(X)
for any 0 6 <9(X)
imply the equivalence of the condition P e %(X) and the condition that P has
the form (1.7). Hence a) is equivalent to c) because a*(P)= X ck0(Q9 x)sk.
k=0
'
Q.E.D.
Remark 1.5. Suppose a differential operator P has R.S. in the weak sense.
Then if we put t = t'k for a sufficiently large fc, then P has R.S. in the coordinate
system (*', x). In fact if P is of the form (1.3), then
Hence P is of the form (1.6) and a(x, s) changes into a( x, -j- J by the transformation (r, x)=(r /fc , x).
Example 1.6. The differential operator
has R.S. along the hypersurface defined by y = 0. Its indicial polynomial
equals 5(5 — l) + -j- — I2 = (s — — + /lYs-- — — 1J and the characteristic exponents
are -y — A and — + A. The operator
has R.S. in the weak sense. If we put t = y2, then P = Q. On the other hand
the operator
has R.S. along the hypersurface defined by x2+yz=l. Then the coordinate
transformation
2j
changes P into R.
y
> l-*2-y\
DEFINITION OF BOUNDARY VALUES
§ 2.
1209
Operations on the Space of Holomorphic Functions
Retain the notation in Section 1. Then the space 0(X) of holomorphic
functions is a Frechet space by the topology of uniform convergences on compact
subsets of X and the element of 0(X) defines a continuous linear transformation
on 0(X).
In general for positive integers N and N' and a ring .R we denote by M(JV,
N' ; jR) the space of matrices of size NxN' whose components are in R and for
simplicity M(N, N; R) is denoted by M(N; R). If rtJeR for i = l,— , N and
j = !,-••, N', then (rlV) denotes the matrix whose (/, j)-component equals rtj.
Under this notation the map
M(JV, N' ; 0ypO)
> M(N, N' ; 0( Y)0])
UJ
^:|
01
^ (/ -. ,(Py))
( n \\
/(Py)
^ P \ i1
ffl
L
is also denoted by a*. For K, Le R+ we put
f^T\^^
)
and
L
K
\t\
where |x| = \x±\ + ••• + |xn|. We identify VL with the submanifold of UKiL defined
by z = 0. Then we can state the main theorem in this section.
Theorem 2.1. Let X be a domain in C1+n containing the closure of U1}L
and Y a submanifold of X defined by f = 0. We define the following conditions
for a matrix P = (PV) in M(N; ®y(X)):
a) There exist non-negative integers m- and m"- such that
N
N
b) // we put m= £ mj+ £ m"h the determinant del (<7*(P)) of the
i=i
j=i
matrix at(P) is a polynomial of degree m for any x in Y.
c) Let Si(x),-~9 sm(x) be the solutions of the equation det (o-.5!(P)) = 0. Then
(A.I)
5v(x)^7V={0, 1, 2,.--}
for any
xeVL
and v = l,— , m.
If the matrix P satisfies the above three conditions, there exists a positive number
K0 g; 1 such that the linear map
(2.1)
1210
TOSHIO OSHIMA
is a surjective homeomorphism for any positive number K which is larger than
K0. Here &(UKtI)N denotes the linear space formed by column vectors of
length N whose elements belong to @(UK}L).
Before the proof of the theorem we give a remark which will be easily seen
by our proof. It will not be referred later and the precise argument is left to
reader's exercise.
Remark 2.2. 1) The number K0 in the theorem depends only on
det (trm/+m^(Pfj-)) and moreover the assumption that the coefficients of Ptj of
order less than m- + m"j are holomorphic only on UKtL in place of X is sufficient
to assure that the map (2.1) is a surjective homeomorphism.
2) Suppose that we omit the condition c) in the theorem. Then if we put
r = max {0} U ({1 + sv(x) | x6 VL and v = I,---, m} n N),
P induces the isomorphism
tr@(UK,i)N ^"> tr@(UKtl)N •
(2.2)
(Here the condition q) equals to the condition r = 0). Hence we can naturally
define the map
P: 0(UK^NltrO(UKtI)N
(2.3)
> @(UKtl
and we have
(2.4)
KerP^KerP,
and
(2.5)
CokerPI^CokerP
00
" J=0
Proof of Theorem 2.1.
(2.6)
Put
P(t, x; 0, tDx)=A(x, 0) + tR(t, x; 0, Dx).
Let 0 and Ox be the rings of formal power series of (t, x) and that of x, respectively. Then P defines the map
P: (9N
(2.7)
> @N.
For u in (9N we put
00
l +n
(i,a)eN
where
*'*
i=0
DEFINITION OF BOUNDARY VALUES
1211
aeJV"
Then for any leNwe have
Pu = (A(x, 0) + tR(t, x; 0, D
»=o
00
°°
__ y ^4fj£ (50w-(x)J*-Iy tR(t x' 0 D }u-(x}t^
l
1=0
'
f=o
1
. j-i
- 2 -4(^5 6>)w i (x)r+ £ tR(t, x; 0, Dx)Ui(x)tl mod
i=J
i=0
and hence
1-1
Here we remark that the condition (A.I) implies
(2.9)
A(x, 1) is invertible in M(N, &(Y))
for any
I e N.
First we want to prove that the map (2.7) is injective. Suppose Pw=0.
Then (2.8) shows
0 = A(x9 0)w0(x) mod tSN,
which means w0 = 0 because of (2.9). By (2.8) and the induction on / we have
and hence (2.9) proves ut = 0. Therefore (2.7) is injective, which assures that
(2.1) is also injective.
Next we want to prove the surjectivity of the map (2.1). Let Atj be the
(z, ^-component of the matrix A( = A(x, (9)) and Atj the cofactor of A^ in A.
Putting A = (Aij), we have
AA = AA = det (A)IN = det (a*(PJ) (x, 0)1N ,
ord Ay ^ m J + m"j and ord ALj g m — m} — mj' ,
where 1N is the identity matrix of size N.
then P' e M(N; &Y(X)) and
Hence if we put Pf ( = (P/ij)) = PA,
and
ord P- j ^ max (ord P£v + ord Avj)
V
^max (ra- + m" + m — m} — m'^)
V
= m + m • — m} .
1212
Tosmo OSHIMA
Using the diagonal matrices E(1) and £(2) with the i-th diagonal components
(0 + l)m~mi and ((9 + l)ms respectively, we put
(2.10)
P"
Then
ord P"j g (m -rn'i)+ (m + m\ - m j) + m}
= 2m.
Here we remark that £(1) defines an injective transformation on (PN, which is
clear by the same argument as before. Hence if P" defines a surjective transformation on &(UK>L)N we can conclude that the map (2.1) is surjective. For
simplicity, denoting P" and 2m by P and m, respectively, we may assume the
following:
(2.11)
ordP z7 gm
and
(2.12)
v*(P) = a(x,s)IN
where a(x, s) (e0(7)[s]) is a polynomial of degree m for any xeY. Let
s^x),---, sm(x) be the solutions of a(x, s) = 0. Then they satisfy (A.I).
Let/e^. W e p u t / =
£
/i «*'*' = Z /fr)*' (A.e C^).
(i,a)eiV 1 + » '
i=0
equation
(2.13)
Then the
Pu=f
means
a(x, /
/
(cf. (2.6)-(2.8)). Hence if we inductively define ut(x) by
u^flOc, / ) - 1 - - (
(2.14)
i=o
then w = IX(x)*1' satisfies (2.13). Therefore (2.7) is surjective.
Next assume fe&(UK^N, where K will be determined later. We want to
prove that u in &N which satisfies (2.13) belongs to &(UKtL)N, It is sufficient to
show ue(9(UKtL^d)N for any SeR+, Since /z(jc) is holomorphic in VL and P is
defined on UKtL for any ^T^l, it follows inductively by (2.14) that ut(x) is
A.
r
also holomorphic in VL. Putting ur= £ w£(;t)/* for re^V, ure&(UK L)N and
DEFINITION OF BOUNDARY VALUES
1213
f-Pur = P(u-ur)etr(PN n V(UKiL)N = tr&(UKiL)N.
Hence replacing / by /— Pur and u by u — wr, we have only to prove that there
exists a positive integer r such that u e @(UK>L-2d)N if /e tr^(^K,L)N- To prove it
by the method of majorant we prepare
Lemma 2.3. 1) Any function in 0(VK >L) can be expressed in a power
series of(t, x) which converges at any point in UK}L.
2) Let 4>(t, x) e 0. Then cj)(t, x) e &(UK}L) if and only if for any L' e R +
satisfying L' <L there exists RL>eR + such that
(2.15)
0(
Here for ^=ZQ, a * f * a and *A' = ZQ,</ xa in ®> ^«^'
majorant series ofij/, that is, C- ja ^|Q >a |/or any i and a.
means that \j/f is a
We continue the proof of the theorem and the proof of the lemma is given
after that. We put
PU = a(x, 0)dtj 4- 1 "Z biM(t, x)0k+
fc=0
£
fc+|a|^m
l«l^l
btjjjtt, x)0*(tDJ* ,
a(x, 0) = a0(x)0m -f di(x)&m~l -\-----h am(x) .
Here we can assume a 0 (x) = l because a0(x) nowhere vanishes on VL. Since
ff
s
m(Pij) i defined on a neighborhood of U ]jL , there exists L' e R such that
Tj-T'
r
L >L and that ffm(Ptj) is defined on t7 lj£ /. Hence putting L" = —^ — , we have
L">Land it follows from Lemma 2.3 that there exists an M 0 >0 satisfying
(2.16)
bijtk.(t9
Similarly we have
1
(fc+|a|<m),
with suitable M5 and ,R5j/ in R+ because bijik and bijikxE&(UKtL),fE tr&(UKiI)N
and a f -e^(F L ). Here K ( ^ l ) and r ( ^ l ) will be determined later and (Rdsf)
1214
TOSHIO OSHIMA
denotes the column vector of length N whose components are Rdif and for $ and
A
\l/ in 0N, <j)»\l/ means that any component of 0 is a majorant series of the
corresponding component of i/r. Moreover (A.I) means
|a(0, OI^CI" 1
(2.18)
(leN)
with Ce R + . Putting
(2.19)
and
(2.20)
PU
we will show that $» w if 0 = l L < ) f * a in ^N satisfies
f c0^»(C0-
(2 21)
\p=(p.j.) and
By (2.14) we have
(2.22)
a(0, / )
U w
=-{/-
with /(/, J8) = {(i,a)6 JV1"1-"!!^/} U {(/, a)e^V 1+w | |a|<|)8|}. Similarly by (2.21) we
have
(2.23)
CPfrf
Now we will prove <^ z ^t z x /? » ulifttlxp by the induction with respect to the lexicographic order on (/, |/?|). If /=0, this is clear because <f>Q,p = uQtp = Q. By
the hypothesis of the induction we have
DEFINITION OF BOUNDARY VALUES
1215
In general if y»i/ and w» w' for y, (/, w and w' in 0, then i? + w»i/ + w', i;w»i/w',
0t? » 0i?' and rZ> Xi [?» f/) v /. Hence by (2.16), (2.17), (2.19) and (2.20) we have
»f(t, x) + (a(0, 0) - P) (
E
(i,a) e ia,/3)
«i,/*a)
and therefore by (2.19), (2.22) and (2.23) we have (j)ltptlx^»ulipt1^.
Thus we
have proved $»w.
Next we will construct a solution 0 of (2.21) with the form (j> = (v)9 where (v)
denotes the column vector of length JV whose components are the same v in 0.
We assume i? = w(z)tr with a formal power series w of one variable satisfying w » 0.
Then
ys.
0D = (@w)tr + w(0tr)y> rwf = rv » v
and
Let M be a positive number which is larger than the number of the elements of
the set {(fc, a)eJV 1 + w |/c4-|a|^m, |a|^l|. Then
(C0» - P)(t?) = " M
+MMSN(I- £
«g0m(v)
with
Now we choose K0, X and r such that
1216
TOSHIO OSHIMA
and put
Then h is holomorphic in £/ KjL _ 2<5 because \g(z)\<C if |z|<L— 25.
putting
Moreover
we have h » y, (9mi; = h and
f )"
^^
This means (fr = (v) is a solution of (2.21) and therefore u«(v). Since
H-----hx H ) converges for (r, x)e UKtL-2s an^ since ^ 7>>t; and (v)»u, u also converges there and therefore u is holomorphic in t/ XjL _ 2 «5.
Thus we have proved that (2.1) is bijective. Since the linear map (2.1) is
continuous and since @(UKiL)N is a Frechet space, it follows from the open
mapping theorem that the map is a homeomorphism.
Q. E. D.
Proof of Lemma 2.3. Putting xn+ L —Kt and replacing n + 1 by n, we may
replace UK^L by F L , (/)(t, ,v) by 0(x) and Kf + XjH ----- hx H by x t H ----- (-X H in the
proof of the lemma.
1) Let 06d?(F L )andxeK L . Put ci = \xi\+~
cf > |xf| and ct H-----h cn < L.
for i = l,-,
n. Then
Hence by Cauchy 's integral formula we have
i| =
where a+l = (acj + !,•••, aw + l).
Since the sum of the integrand converges
DEFINITION OF BOUNDARY VALUES
1217
uniformly on the pathes of the integration,
2) Let Le/? + and 0= E a a - x « e x . For any L'eR+ with L <L we
aelV"
assume the existence of RL>eR+ such that $«Run- *i + "; + *" ^ . Let
x G FL and put L' = JM+A and fca = #L, • Jffi g| . Since
we have
£ |«axa|^ £ 5a|x|a = ^
L
l - - - ) X <cx).
Hence
the series
* converges.
On the contrary assume 0 converges for any x e FL. Let L e R+ with L'
r ' _i_ r
<Land put L"' = •— y-— . Then L' <L" <L and Sl fl a^ a l converges uniformly
on VL» and defines a continuous function. Let ML> be its maximal value on
VL». Then |aaca|g ZK^I^M^ for any c = (c 1 ,---, c,,)e [0, L"]« satisfying
Cj 4- • • • + cn = L" . If the relation
(2.25)
l f l f
«
a!
«
is valid, then we have
and hence
Z fl,
L'
with RL> = max ML,(i +1)""1 (-JTT)
(< oo).
We will show (2.25). It is clear when a = 0. Therefore we may assume
oc^O. We define ce Rl such that ^i = ^l- (resp. c,- = 0) if oe^O (resp. af = 0).
Let p E N" with |p\ = |a|. If a r -^ft, then
a,!
1218
Tosmo OSHIMA
If &i<Pi, then
a,!
Hence
and
, ,
hCw)lal =
£"|a| = ( C l H
l/?lt
X
laM
I P I - c/?^(|a|4-])n-l. l ^ l -
I0?=l«l
because 9{peN"\ |]8| = |a|}^(|a| + l)11--1. Thus we have (2.25).
Q.E.D.
For the operators in &f(X) we have the following theorem:
Theorem 2.4. Let X be a domain in C1+n containing the origin and Y a
submanifold defined by t = Q. Let P = (P^(£, x; 0, DJ) be a matrix in
M(N; @$(X)) and r be a positive integer such that P//P, x; — , DXJ has R.S.
along Yfor z, j = l,---, N (cf. Remark 1.5). For K and Lin R+ we put
Let L be a positive real number such that X contains the closure of Url}L.
We assume P satisfies a) and b) in Theorem 2.1. Let s^x),---, sm(x) be the roots
of det (o\|.(P)) = 0. Here we assume
(A.I)'
rsv(x)£N
in place o/(A.l).
for
v = l,-,m
Then there exists K0 6 R+ such that the map
is a surjective homeomorphism for any K satisfying K>KQ.
Proof.
a
For any u =
M, ar x and w* =
£
X
(i,a)6JV 1 + "
(i,a)e2V 1+ »
",- a r*x a in 0N(ui a e CN), we put u* =
wri Jx*.
'
'
Moreover we put P*=pitr, x ; — ,
'
\
\
r
Since P^ satisfies the assumptions in Theorem 2.1, there exists K0 ^ 1 such that if
then the map P*: &(UKtL)N-»<!)(UKiI)N is bijective.
We choose KQ and K as above and define the linear maps
)W and
V: 0(Cfc. L ) w —
UJ
Z/ I
DEFINITION OF BOUNDARY VALUES
Let IN be the identity map on 0(UKtL)N.
1219
Then
(/*-*W(l/JW.)"), P^^(G(VK^)^WO(UK^ and
c=(/N — ^^)((P(UK>L)N).
Therefore since the map P* is a bijective transformation
on &(UKtI)N, it also defines a bijective transformation on $'F(&(UK}L)N) =
$(&(UK,L)N)- Moreover since *F3> is the identity transformation on ^(UrKiL)N.l
the map P=WP*<I> is a linear bijective transformation on &(UrK>L)N. Hence P
is a homeomorphism because of the open mapping theorem.
Q. E. D.
The following theorem is fundamental in the theory of differential equations.
It is an easy corollary of Theorem 2.1 but will not be referred later.
Theorem 2.5 (Cauchy-Kowalevsky). Lei Q(t, x: £>„ Dx) be a differential
operator of order m defined in a neighborhood of Ui}L. Assume (7m(P)(0, x, 1,
0)^0 for any xeVL. Then there exists a positive number K0 such that the
linear map
(2.26)
0(UKtL) - > 0(UKiL)@ ("0 0(VL))
i=0
u I-
> (G/; />?/lr=o,-, ^-yit-o)
is a surjective homeomorphism for any K>K0.
Proof.
Put Q = %qitU (t, x)D\D* and P = Qtm. Since
P has R.S. along 7 and cr#(P) = #m,0(05 x)(s4-l)---(s + m). Since the characteristic exponents of P are — 1, — 2,---, — m, the operator P satisfies the assumptions in Theorem 2.1 and hence there exists a positive number K0 such that
the map
(2.27)
P = g*»: 0(VKJ ^
0(17^
is a bijection for any K>K0.
We indentify 0(VL) with a subspace of &(UKtL) by the coordinate system
(f, x). For any (/; u 0 ,"' 5 ^m-i) e < ^(^L)©( © ^(VLJ), there exists a function
i=0
w e @(UK^L) which satisfies
because (2.27) is surjective. Putting u = tmw + v0 + - - V i + •••+.
1220
Tosrao OSHIMA
we have (Qti; D?u| f = 0 ,---, /)f~ 1 wUo) = (/; «W> y m-i)- This means (2.26) is
surjective.
Let ue&(UKtL) such that DJM|, =0 = 0 for / = (),•••, m-1. Then w = Pw
with a function we^l/^J. Suppose Qw = 0. Then Qtmw = 0 and (2.27)
proves w = 0. Hence (2.26) is bijective.
Thus we have proved that (2.26) is a continuous bijective linear map and
hence (2.26) is a homeomorphism.
Q. E. D.
Corollary 2.6. Let P be a differential operator of order m defined in a
neighborhood of UltL. Assume P has R.S. along VL in the weak sense. Let
Si(x),'-, sm(x) be its characteristic exponents. Assume moreover s^xJ^Q and
rsv(x)$£Z+ for any xeVL and v = 2,---,m under the notation in Theorem 2.4.
Then we have the bijections
(2.28)
0(UiiL) 2^tO(U'KtL) @ 6(VL}
U)
u
I-
,-
UJ
» (PW? u\t=Q)
and
(2.29)
{uee>(UitL)\Pu=0} ^
&(VL)
> u
t=0.
Proof. Under the expression (1.3), a0 = 0 because 51(x) = 0. Hence P
= tQ with a differential operator Q. Since Qt = r1Pt1=P(t, x; 0+1, Dx), Qt
has R.S. in the weak sense and any of its characteristic exponents never take
values in N for any xeF L . Therefore Theorem 2.4 implies Qt:
Note that if u e &(UrKiL), then Pu = tQu e t(9(UrK>L). For arbitrary functions
fe&(UrKiL) and ve0(VL), we have (Qt)w=f-Qv with we^(l/^ L ). Hence
P(rw + t;) = ?/ and (tw + v)\t=0 = v. On the other hand if a function M e &(UrKtL)
satisfies M| r = 0 = 0 and P« = 0. Then u = tw with we&(UrKtL) and tQtw = Q.
Hence w = 0 and therefore w = 0. Thus we have proved (2.28) is bijective and
hence (2.29) is also bijective.
Q. E. D.
Now we consider the following situation: There exist manifolds Q, X'
and Y' of dimension r, l + nf and n'9 respectively, such that Y= Qx Y' and X = Q
xX'. Then n = r + ri. Let A = (A 1 ,--- 5 /lr), z = (z 1 ,---, ZB/) and (^, z) = (/, Zj,---,
z/r) be the coordinate systems of Q, Y' and X', respectively, such that Y' is defined
by t = 0. A holomorphic differential operator P on X is said to be a differential
DEFINITION OF BOUNDARY VALUES
1221
operator on X' with a holomorphic parameter A if P is of the form P = P(A, t, z;
Dt9 Dz), that is,
Now suppose that P(A, f, z; Df, Dz) has regular singularities along Y and assume
the conditions:
(A.2)
Any characteristic exponent sv(A, z) of P does not depend on z.
(A.3)
sv(A, z)- v(A, z)£ Z
for any (A, z) e 7 and v ^ v'.
Then consider the equation
(2.30)
Pw=0.
Definition 2.7. The solution u of the equation (2.30) is called an ideally
analytic solution if u is of the form
m
O ji)
^1^
v^.
11— V
n (I
t -V*v(A)
14—
/ „M
V ^A, t, z.yi
v=l
with holomorphic functions #V(A, f, z) defined on a neighborhood of Y.
Corollary 2.8. We can choose a neighborhood U of Y satisfying the
following: For any (fr^A, z),---, frw(A, z))e0(Y) m f/iere ex/sfs a unique
(a^A, r, z),-", tfm(A, ?, z))e^(C/) w swc/i that the function u given by (2.31) is a
solution o/(2.30) with the condition 0V(A, 0, z) = b v (A, z)/or v = !,•-•, m.
Proo/. Put Qv = t-*v(VptsvW for v = l,-», m. Since P is of the form
P(A, t, z; 0, tDz), QV = P(A, f, z; 6) + sv(A), tDz). Hence Qv has R.S. along Y
and its characteristic exponents are
sM(A)-sv(A) (/i = !,-••, m).
Therefore Qv satisfies the assumption in Corollary 2.6 and there exists a unique
function #V(A, f, z) in 0(L7) such that P v a v = 0 and that av(A, 0, z) = frv(A, z).
Here 17 is a suitable neighborhood of Y. The function u defined by (2.31) is
clearly a desired solution of (2.30).
Let M be a function of the form (2.31). Since
v=l
Sl(A
Sm(A)
V
v=l
and * V--, ?
are linearly independent over 0(17), the condition Pw = 0
implies Q v a v = 0. Thus we have the uniqueness of the solution by Corollary 2.6.
Q. E. D.
1222
TOSHIO OSHIMA
§ 3.
Definition of Boundary Values
Let M be a (1 + w)-dimensional real analytic manifold and N an n-dimensional submanifold of M. Assume M is devided by N into two connected
components M+ and M_. We choose a local coordinate system (t, x) =
(t, *!,-••, xn) °f M so that N, M+ and M_ are defined by * = 0, *>0 and t<0,
respectively. Let Q be a domain in Cr. Let P be a differential operator on M
with a holomorphic parameter /leO. We assume that there exist a complex
neighborhood X of M and a complexification 7 of N in Z such that P can be
extended to a holomorphic differential operator on X with a holomorphic
parameter A e O. For simplicity we identify P and the extension. We moreover
assume that if we regard P as a differential operator on Q x X, P has R.S. along
Q x Y. In this case a differential operator P on M is said to have R.S. along N.
We assume that any characteristic exponents of P does not depend on x but
depends holomorphically on A e Q. We denote the order of P by m and the
characteristic exponents by s^A),---, sm(A).
For a real analytic manifold 17, let jf(U) (resp. ff°°(I7), #'(*/) and ^(£7))
denote the linear space of all analytic functions (resp. infinitely differentiable
functions, Schwartz's distributions and Sato's hyperfunctions) on U. If & is
jaf, ^f00, ^' or # and Wis a subset of [/, then we set
| supp/is compact} .
Let n&(M) denote the linear space of all hyperfunctions on M with a holomorphic parameter A 6 O, that is,
, with
In this section we always use the above notation and assume that P is the
above differential operator. Then the following theorem is essential to define
the boundary values of the solutions of the equation Pw = 0.
Theorem 3.1.
Put
DEFINITION OF BOUNDARY VALUES
1223
If any characteristic exponent of P does not take the value in negative integers,
then the restriction maps
(3.1)
tp
:
U
(3.2)
U
fl^"[Af+]r^fl^"
are bijective.
To prove the theorem we prepare :
Lemma 3.2. Under the same assumption as in Theorem 3.1, the following
maps are bijective,
P: ^ 0 x M [^ x TV] 2^ # fl x M [O x ^1
(3.3)
U
Proof.
(3.5)
U
#oxAf [0 x N] Cx ^0xM[Q x #1-
(3.4)
Let P* be the adjoint operator of P.
P = a(A, x, 0) + fS(M, *;6» +
E
fe+|a|^m
Put
rM(A, t, x)©^*)* .
|a|>0
Then 0(A, x, s) is the indicial polynomial of P. Since <9* = (tDr)* = — <9 — 1,
P* has also R.S. along AT whose indicial polynomial equals a(A, x, — s — 1).
Therefore the characteristic exponents of P* are — 1 — s^/l),---, — I — sm(/l).
The assumption implies that any of them does not take the value in non-negative
integers. Hence for any compact subset K of Q x N, we have the topological
isomorphism
(3.6)
P* : ^^
by Theorem 2.2. Here the space J^OXM^) has the natural topology induced
by the inductive limit of the Frechet spaces jtf(U), where U runs over fundamental
neighborhoods of K in a complexification of O x M. Then the topological dual
of the map (3.6) means
(3.7)
P: ^X
Now consider the map (3.3). Let / be a function in &Q*M[Q x JV]. Since
the sheaf of hyperfunctions is flabby, / can be expressed as a locally finite sum
/=£/£, where /j6^ xM [^2x JV] and the support of ft is compact. Then the
map (3.7) implies that there exist #£e,^xM[ su PP/i] satisfying Pgr f =/ f . Since
^Z0i=/> we see that the map (3.4) is surjective. Suppose P/=0 to prove the
injectivity of the map. For any point p of Q x AT there exists a neighborhood V
1224
TOSHIO OSHIMA
of p such that the set / = {i|(suppt/;.) n V^cj)} is finite. Put /'= E/r
Then it
iel
SU
follows from (3.7) that there exists h' e ^XMC PP Pf\ which satisfies Ph' = Pf.
The injectivity of (3.7) means f' = h' and hence / equals 0 in the neighborhood
V of p because f\v=ft\v = h'\v and suppP/| F = 0. Thus we have proved that
the map (3.3) is bijective.
Next consider the case of distributions. We can prove that the map (3.4)
is bijective by the same way as in the case of hyperfunctions but we use a more
explicit method. Let V be any relatively compact open subset of Q x M. Then
any element /e ^ X M[^ x N] is locally of the form
(3.8)
f\v=,
i=0
with distributions /f(A, x). Here <5 (i) (f) is the i-th derivative of the Dirac's
^-function of the variable t. Using the expression (3.5) we have
(3.9)
P/;(A, x>5«>(0 =fl(A,x, _ j - i)/.(A, x)«5(0(0 + '£ g^
x ^(v )(t)
v=0
with suitable distributions gv(h, x). We note that the assumption says that
a(A, x, — i — 1) does not vanish at any point in Q x N. Hence considering (3.8)
and (3.9), we can easily prove by the induction on / that the restriction of the
map (3.4) on Fis both injective and surjective. This proves that the map (3.4)
is bijective.
Q. E. D.
Proof of Theorem 3.1. Let /e 8(Q x M +) which satisfies P/=0. In view of
the flabbiness of the sheaf of hyperfunctions there exists a function g e 0$(Q x M)
such that g\n*M+=f an^ 01r2xM_ = 0- Since Pge&QxM[QxN~\, Lemma 3.2
proves the existence of /ze^ f l x M [Ox JV] with Pg = Ph. Replacing g by g — h,
we may moreover assume the condition Pg = 0. Thus we have proved that the
restriction map
(3.10)
*
is surjective, where ^^ x M [OxM + ] = {w e&QxM[QxM +~]\Pu = Q] and
&p(QxM+) = {ue@(QxM+)\Pu = Q}. The injectivity of the map (3.10) also
follows from the same property of (3.3).
Under the above notation we assume fe Q^P(M+). Then P(d^.g) = 5^(Pg)
= 0 and supp B^g c Q x N for i = l,---, r. Hence the injectivity of (3.3) implies
3 A .0=0. Thus we can conclude that the map (3.1) is bijective.
In the same way as above we can prove that the map (3.2) is bijective if we
DEFINITION OF BOUNDARY VALUES
1225
note that for any f£&'(QxM)\QxM+ there exists an extension ge@'(QxM)
such that g\QxM+ =f and that supp/c£ x M+ (cf. [2]).
Q. E. D.
Now we will define boundary values of the solutions of the equation
(3.11)
P(A, f, x ; 0 , fD x )w=0,
where the operator P satisfies the assumption mentioned in the first part of this
section. We pay attention to one characteristic exponent of P, say s/(A), and
assume
(3.12),
s f U)~s v (A)^Z +
for any AeO
and v = l , • • - , m.
P
Let w(A, t, x) be a function in 0^ (M+), that is, H is a hyperfunction solution of
the equation (3.11) which is defined on M+ and has the holomorphic parameter
AeO.
There exists a differential operator Q^(A, r, x; Df, Dx) which satisfies
(3.13)
rs^Pts^=tQi.
In fact, since sf(A) is a characteristic exponent of P, P is of the form P = (<9 — Sj(A))6(A, x, 0) + fjB(A, r, x; 0, DY) and therefore Q, = D,ft(A, x, 0 + s,.(A))-f-£(A, t, x;
0 + sf(A), DJ. Here we remark that the operator p. = fQ. has R.S. along JV and
its characteristic exponents are sv(A) — s,(A) (v = l,•••, m). Moreover we have
p.^-si(A) l/ = ? -si(A)p M _Q Then in view of Theorem 3.1 we have a unique wf
6 fi ^ Pi [M + ] such that w f | 0 x M + = f~ Sf(;i) w because (3.12)f assures the assumption
in the Theorem 3.1. Since tgjWj = 0, we have
(3.14)
with 0;(A, x) e Q&(N).
That is,
(3.15)
where £P. is the map defined in Theorem 3.1.
Definition 3.3. The function (^(A, x) e Q&(N) defined as above is called
the boundary value of the solution w(A, t, x)e fi ^ p (M + ) with respect to the
characteristic exponent s/(A).
Remark 3.4. If an equation Q(t, x; Dt, Dx)u(t, x) = 0 of order m is noncharacteristic with respect to N, then
w-l
i=0
1226
Tosfflo OSHIMA
with $i(x)e&(N).
Then ($0(;c),---, 0m-i(x)) is called the boundary value of
u(t, x) by Komatsu-Kawai [6] and Schapira [9].
Now we consider the boundary values of the distribution solutions or the
ideally analytic solutions:
Theorem 3.5. Retain the notation in Definition 3.3.
1) Ifuea9'*(M+)<\&(QxM)\QxM+9
then &(A, x)e09'(N).
2) Let 0(A, x, s) be the indicial polynomial of P. If
(3.17)
s£(A) - s/A) <£ Z
/or flr/ij; /, j = 1 , - • • , m 0/w/ A e &
M is an ideally analytic solution of the form
(3.18)
namely /V(A, (, x) are analytic in a neighborhood ofQxN, then
(3.19)
0A *) =
, 0,
\
s=0
Proo/. 1) If u e®'(Q x M)| flxM+ , then r-«*>ii e ^'(Q x M)| flxM+ (cf. [2]).
Hence the first part of the theorem is a direct consequence of Theorem 3.1 and the
definition of the boundary value.
2) Since the map of taking the boundary values is C-linear, we have only
to determine the boundary value of the solution /v(/l, t, x)tSvW with respect to
the characteristic exponent sv(A) (¥ = !,•••, m). Put w£=/v(A, t, x)^ v(A) ~ 5f(A) .
Here t*+ is a distribution-valued meromorphic function of s whose poles are
negative integers (cf. [1], [8]).
In general for analytic functions /(A, t, x) and s(/l), where s(A) is never equal
to any negative integer,
, t, xy)+s(X)f)&\
)t^\
Hence if Q is a differential operator of the form g(A, t, x; 0, Dx) and
t, x)tsW)=g(A, t, x)t*W on QxM+ with a function g, then g(/(A5 r,
= ^(A, t, x)4a>.
Since /V(A, r, x)t Sv(A) ~ Si(A) is a solution of the equation
(3.20)
(
so the function w£. Hence ^^^(r^^f^t, A,
On the other hand, since a(A, x, 5£(A)) = 0,
DEFINITION OF BOUNDARY VALUES
(3.21)
1227
a(A, x, 5 + sf(A)) = sfo(A, x, s)
with a polynomial &(A, x, s) of s. Then
with a differential operator R(A, t, x; (9, Dv), which implies
(3.22)
Qi = Dtb(^ *, 6>) + #(A, r, x; (9, D,).
Consider the case where v ^ i. It follows from this expression that
=WA, t, x)^w-.i(A)-i
with an analytic function ^v. Since sv(A) — Si(A) — 1 does not take the value in
negative integers and since tQtUi=Q, we have i^v = 0 and therefore the boundary
value is zero.
Next consider the case where v = f. Here we remark that f£ coincides with
the Heaviside's function Y(t). Putting
/f(A, t, x)=/ f (A, 0, x) + g&9 t, x)t,
we have
t(l,
0, x)
, 0, x)Y(t)) + (D,fc + K)(0A t, x)t+)
=fc(A,x, 0)/A 0, xWO + ^a t, x)7(0
with an analytic function h^ Since Q = tQiui = hi(h9 t, x)t+9 we can conclude that
ht = 0 and that the boundary value equals Z?(A, x, 0)/f(A, 0, x).
Q. E. D.
The following theorem relates to the induced equations on the boundary.
Theorem 3.6. Retain the notation in Definition 3.3. Let R^A, t,x',0, Dx),
•••,Ri(h t, x; 0, Dx) be differential operators which satisfy the following
conditions:
a) £/A, t, x; 0, £>>(A, t, x) = Qfor j = l9-9 I.
b) There exist differential operators S) of the form S)(A, t, x; 0, tDx)
0", k = !,•••, 0 such that
(3.23)
[P,*y]=
(3.24)
ord S) < ord P + ord Rj - ord Rk ,
fc=i
1228
(3.25)
TOSHIO OSHIMA
ff*(S))
=0
for j,k=l,-,l.
Then the boundary value (/>f(A, x) satisfies
(3.26)
Rj(^ 0,x; S l <A), DJ0A x)=0
for 7 = 1,-, /•
Proof. Let J? denote the column vector of length / whose k-ih component
equals Rk and let S denotes the square matrix of size I whose (j, /c)-component
equals S). Then the assumption says
(3.27)
RP = (P + S)R,
where P is identified with the scalar matrix whose diagonal components equal P.
We will retain the same notation which was used to define the boundary value.
Put p' = r8*WPtsiW, S' = r8'WSts*W and Rf = rs^Rts^. Then (P' + S7>
R'ui = R'P'tii = RftQini = Q and R'ui\QxM+ = rSi^}Ru = 0. On the other hand,
using Theorem 2.1, we can prove that the map
(3.28)
P' + S': 0a*MlQxNll - >^ x M [OxAT]>
is bijective in the same way as in the proof of Lemma 3.2 because ff*(Pf 4-S") is a
scalar matrix whose y-th component equals (7*(P)(A, x, s -f- Sj(A)). Thus we can
conclude that R'ut = Q.
Now since cr^S") = 0, we have S' = tT' with a suitable matrix T' of differential operators and therefore (t~1Rft)Qiui = (Qi+T')R'ui = Q, which means
, 0, x; sJW, DJ0A x)d(t) = Rj(l, t, x;
=0
for j = !,•••, /. Thus we have the theorem.
Q.E. D.
The following theorem is used to define the boundary value globally on a
manifold.
Theorem 3.7. Let u be a function in Q&P(M+) and use the notation in
Definition 3.3.
1) The Q&(N)-vahied section <^(;., x)(df) s ' (A) of (TjjfM)® 5 '^) is independent of the choice of local coordinate systems. Namely, let (tr, xf) be
another local coordinate system of M which satisfies t' = c(t, x)t and xfj =
Xj(t, x) (j=l,---, n} with c(t, x)>0 and let 4>'i(k9 xr) be the boundary value of u
with respect to characteristic exponent s£(A) which is defined by using the coordinate system (tf, x'). Then
(3.29)
&(A, x) = 0j(A, x'(0, x)) (c(0, x))s«'« ,
DEFINITION OF BOUNDARY VALUES
1229
2) Let F^k, t, x) and F2(A, f, x) be real analytic functions on M which
never vanish anywhere and let jR(A, t, x; (9, Dx) be a differential operator with
R = ^AjRJ.> where Rj are the differential operators in Theorem 3.6 and Aj
are suitable differential operators of the form ^4/A, t, x; 0, Dx). Let ^"(/l, x)
be the boundary value ofF2u which is defined by using (FlP + tR)F2l in place of
P. Then #(A, x) = F&, 0, x)<^(A, x).
Proof. 1) Let Q\ be the differential operator defined by the coordinate
system (t1, x') which corresponds to Qt. Then
= c(t,
= c(t,
Hence
(3.30)
Q'i = c(
Since
t'
and since
f'QiWf,
we have
^'
and therefore
Thus by Definition 3.3 we have
which is equivalent to (3.29).
2) It follows from the proof of Theorem 3.6 that (t~Si^\Fi
F2ui = F1tQiui+ £ L4/A, t, x; 6) + s£(A), D^u^O.
Since F2i1^xM+ = r s '< A >F 2 w, We have
1230
TOSHIO OSHIMA
&, *, x;
= F1(A, t,
which implies the theorem.
Q. E. D.
References
[ 1 ] Gel'fand, I. M. and Shilov, G. E., Generalized Functions, Academic Press, New York,
1964.
[2] Lojasiewicz, S., Sur la probleme de la division, Studia Mathematica, 17 (1959),
87-136.
[ 3 ] Helgason, S., A duality for symmetric spaces with applications to group representations,
Advances in Math., 5 (1970), 1-154.
[ 4 ] Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T. and Tanaka, M.,
Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math.,
107 (1978), 1-39.
[ 5 ] Kashiwara, M. and Oshima, T., Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math., 106 (1977), 154-200.
[ 6 ] Komatsu, H. and Kawai, T., Boundary values of hyperfunction solutions of linear
partial differential equations, Publ RIMS, Kyoto Univ., 7 (1971), 95-104.
[ 7 ] Oshima, T. and Sekiguchi, J., Eigenspaces of invariant differential operators on an
affine symmetric space, Inv. Math., 37 (1980), 1-81.
[ 8 ] Sato, M., Theory of hyperfunctions, I, II, /. Fac. Sci. Univ. Tokyo, Sect. I, 8 (1959),
139-193,357-437.
[ 9 ] Schapira, P., Probleme de Dirichlet et solutions des hyperfonctions des equations
elliptiques, Bull. UMI (4), 3 (1969), 369-372.